\subsection*{Please Take Note!}

We will be covering everything relevant here, and when the time comes we will
be performing in excrutiating detail every Feynman diagram of significance in
QED.

\input{em}

\section{A Comically Brief Review}
The following is a table summarizing the properties of the solutions of the 
Dirac equation.

\begin{tabular}{|p{3cm}|c|c|}
\hline
Property & Electrons & Positrons\\ \hline
Spinor components & $\psi(x)=au^{(s)}(p)\exp[-(i/\hbar)p\cdot x]$ & $\psi(x)=av^{(s)}(p)\exp[-(i/\hbar)p\cdot x]$\\ \hline
Momentum Space Dirac Equation & $(\gamma^\mu p_\mu - mc)u = 0$ & $(\gamma^\mu p_\mu + mc)v = 0$ \\ \hline
Adjoint Dirac Equation & $\bar{u}(\gamma^\mu p_\mu - mc) = 0$ & $\bar{v}(\gamma^\mu p_\mu + mc) = 0$ \\ \hline
Orthogonality & $\bar{u}^{(1)}u^{(2)} = 0$ & $\bar{v}^{(1)}v^{(2)} = 0$ \\ \hline
Normalization & $\bar{u}u = 2mc$ & $\bar{v}v = -2mc$ \\ \hline
Complete & $\sum_{s} u^{(s)}\bar{u}^{(s)} = (\gamma^\mu p_\mu + mc)$ & $\sum_{s} v^{(s)}\bar{v}^{(s)} = (\gamma^\mu p_\mu - mc)$ \\ \hline
\end{tabular}

A free photon, on the other hand, of momentum $p = (E/c, \bold{p})$ with
$E = |\bold{p}|c$ is represented by the wave function
\begin{equation}
A^{\mu}(x) = ae^{-(i/\hbar)p\cdot x}\epsilon^\mu_{(s)}
\end{equation}
where $\epsilon^\mu$ is a spin dependent vector, $s=1,2$ for the two polarizations
(``spin states'') of the photon. The polarization vectors $\epsilon^\mu_{(s)}$
satisfy the \emph{momentum space Lorentz condition:}
\begin{equation}
\epsilon^\mu p_\mu = 0.
\end{equation}
 They are orthogonal in the sense that 
\begin{equation}
\epsilon_{\mu(1)}^{*}\epsilon^{\mu}_{(2)} = 0.
\end{equation}
They are further normalized
\begin{equation}
\epsilon^{*}_{\mu}\epsilon^{\mu} = 1.
\end{equation}
In the Coulomb gauge
\begin{equation}
\epsilon^0=0,\quad \epsilon\cdot p = 0
\end{equation}
and the polarization three-vectors obey the completeness relation
\begin{equation}
\sum_{s=1,2} (\epsilon_{(s)})_i (\epsilon^{*}_{(s)})_j = \delta_{ij} - \hat{p}_{i}\hat{p}_{j}.
\end{equation}

\section{Quantum Electrodynamics}
\input{rulesQED}
\section{Elastic Processes}

An elastic (relativistic) process is one where kinetic energy, rest energy, and
mass are all conserved. We will explore such examples in QED.

\input{QEDexOne}
\input{moller}

%
% \section{Inelastic Processes}
% An inelastic (relativistic) process is one where kinetic energy, rest energy,
% or mass are not conserved. We will explore such examples in QED.

% % Anamolous magnetic moment for the electron
% \input{thirdOrderEx}
%
